Kohn-Luttinger superconductivity on two orbital honeycomb lattice

Motivated by experiments on twisted bilayer graphene, we study the emergence of superconductivity from $\textit{weak}$ repulsive interactions in the Hubbard model on a honeycomb lattice, with both spin and orbital degeneracies, and with the filling treated as a tunable control parameter. The attraction is generated through the Kohn-Luttinger mechanism. We find, similar to old studies of single layer graphene, that the leading superconducting instability is in a $d$-wave pairing channel close to Van Hove filling, and is in an $f$-wave pairing channel away from Van Hove filling. The $d$-wave pairing instability further has a twelve-fold degeneracy while the $f$-wave pairing instability has a ten-fold degeneracy. We analyze the symmetry breaking perturbations to this model. Combining this with a Ginzburg-Landau analysis, we conclude that close to Van Hove filling, a spin singlet $d+id$ pairing state should form (consistent with several other investigations of twisted bilayer graphene), whereas away from Van Hove filling we propose an unusual spin and orbital singlet $f$-wave pairing state.